Discrete Time-Scale Analysis

Summary

In Chapter 3, we expressed a signal / as a continuous superposition of a family of wavelets Ψs,t, with the CWT ƒ (s,t) as the coefficient function. In this chapter we discuss discrete constructions of this type. In each case the discrete wavelet family is a subframe of the continuous frame { Ψs,t}, and the discrete coefficient function is a sampled version of ƒ (s,t). The salient feature of discrete wavelet analysis is that the sampling rate is automatically adjusted to the scale. That means that a given signal is sampled by first dividing its frequency spectrum into "bands," quite analogous to musical scales in that corresponding frequencies on adjacent bands are related by a constant ratio δ > 1 (rather than a constant difference v > 0, as is the case for the discrete WFT). Then the signal in each band is sampled at a rate proportional to the frequency scale of that band, so that high-frequency bands get sampled at a higher rate than that of low-frequency bands. Under favorable conditions the signal can be reconstructed from such samples of its CWT as a discrete superposition of reciprocal wavelets.